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On generalised alpha connectedness in bipolar intuitionistic fuzzy topological environment

K.Ludi Jancy Jenifer#1, M.Helen*2 #Department of Mathematics,Nirmala College for Women, Bharathiar University

[email protected] [email protected]

Abstract — The concept of this paper is to bring to spotlight the theory of generalised alpha connectedness in Bipolar intuitionistic fuzzy topological environment thereby studying inter relationships and properties.

Keywords — Bipolar intuitionistic fuzzy connected space, bipolar intuitionistic fuzzy -connectedness, bipolar intuitionistic fuzzy

-extremally disconnected space,bipolar intuitionistic fuzzy super connected ,bipolar intuitionistic fuzzy -super connected

space,bipolar intuitionistic fuzzy -disconnected spaces(i=1,2,3,4).

I. INTRODUCTION

Atanassov[1] generalised the idea of fuzzy sets which was introduced by Zadeh[10] to a new class of intuitionistic fuzzy sets. Later C.L. Chang[2] came with the new concept of fuzzy topological spaces and Dogan Coker[4] gave an introduction to intuitionistic fuzzy topological spaces. Bipolar valued fuzzy sets, which was introduced by Lee[6] in 2000, stands as an extension of fuzzy sets whose membership degree range is enlarged from the interval [0,1] to [-1,1]. In the year 2015,D. Ezhilmaran & K. Sankar[5] ,discussed on the morphism of bipolar intuitionistic fuzzy graphs and developed its related properties.Later bipolar intuitionistic fuzzy sets in a soft environment was put forward by Chiranjibe Jana and Madhumangal Pal[3] in the year 2018.K.Ludi Jancy Jenifer and M.Helen[7] introduced and studied bipolar intuitionistic fuzzy generalised alpha closed sets via bipolar intuitionistic fuzzy topological spaces.K.Ludi Jancy Jenifer and M.Helen[8] discussed bipolar intuitionistic fuzzy generalised alpha continuity and various forms of irresolute functions.Connectedness in intuitionistic fuzzy special topological spaces was introduced by Oscag and Coker[9] during the year 1998. This paper focuses on bipolar intuitionistic fuzzy connected space,bipolar intuitionistic fuzzy connectedeness and various forms of connectedness.Several properties concerning their inter relationships and properties are investigated.

II. PRELIMINARIES

Definition 2.1[5]: Let X be a non empty set.A bipolar intuitionistic fuzzy set B={(x, μP(x),μ N (x), P (x), N (x))| x X} where, μP : X → [0,1], μN : X → [- 1,0] ,γ P : X → [0,1], γ N : X → [- 1,0] are the mappings such that 0 μ P (x) +

P (x) 1 and −1 μ N (x) + N (x) 0.

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Definition2.2[3]:

Let A and B be two bipolar intuitionistic fuzzy sets given by ={< >;x X}

and B={< >;x X >} in the universe X .If ,

, then A B. Definition 2.3[5]: The union of two bipolar intuitionistic fuzzy sets A and B in the universe X written as C= A B which is given by

(A B)(x)={< , , >}

Definition 2.4[5]: The intersection of two bipolar intuitionistic fuzzy sets A and B in the universe X written as C= A B which is given by

(A B)(x)={< , , >}

Definition 2.5[3]: The complement of bipolar intuitionistic fuzzy set A in the universe X written as ,which is given by

={1- ,-1- ,1- ,-1- }

Definition 2.6[3]: Two bipolar intuitionistic fuzzy sets A and B in the universe X is said to be equal and written as A B if and only if A B and B A. Definition 2.7[7]: The bipolar intuitionistic fuzzy empty set may be defined as: Definition 2.8[7]: The bipolar intuitionistic fuzzy absolute set may be defined as : 1 ={<x,1,-1,0,0>:x X} Definition 2.9[7]: A Bipolar intuitionistic fuzzy topology (BIFTS for short) on a nonempty set X is a family of bipolar intuitionistic fuzzy subsets in X satisfying the following axioms Axiom1:

Axiom 2: for any { }

Axiom 3:A B for any

In this case the pair (X, ) is called a bipolar intuitionistic fuzzy topological space(BIFTS for short) and any bipolar intuitionistic fuzzy set(BIFS) in is known as a Bipolar intuitionistic fuzzy open set (BIFOS for short ) in X.The complement of a BIFOS is called a bipolar intuitionistic fuzzy closed set(BIFCS).

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An BIFS X is called bipolar intuitionistic fuzzy clopen (BIF clopen) if and only if it is both bipolar intuitionistic fuzzy open and bipolar intuitionistic fuzzy closed. Definition 2.10[7]: Let (X, ) be a BIFTS and A={<x, } be a BIFTS in X.Then the bipolar intuitionistic fuzzy interior and bipolar intuitionistic fuzzy closure are defined by BIFint(A)= {G/G is anBIFOS in X and G A} BIFcl(A)= {K/K is an BIFCS in X and A K } For any BIFS A in (X, ) we have = BIF int ( ) and = BIFcl ( ) Definition 2.11[8]: Bipolar Intuitionistic fuzzy point: Let , ) (0,1) and ( ) (-1,0) such that + 1 and + -1.A bipolar intuitionstic fuzzy point(BIFP in short) of X is a bipolar intutionistic fuzzy set of X defined by

=<x, , (x)> .For x X,we have

= =

= =

In this case,x is called the support of A bipolar intutionistic fuzzy point is said to belong to a bipolar intuitionistic fuzzy set A=<x, , (x)> of X,defined by c( ) A if

, , (x) . Definition 2.12[8]: Bipolar intuitionistic fuzzy neighbourhood: Let c( ) be a bipolar intuitionistic fuzzy point of a BIFTS(X, ).A BIFS,A of X is called a bipolar intuitionistic fuzzy neighbourhood (BIFN) of c( ) if there is a BIFOS,B in X such that c( ) B A. Definition 2.13[7]: A subset A of a bipolar intuitionistic fuzzy topological space (X, ) is called a bipolar intuitionistic fuzzy generalised alpha closed set[11] if cl(A) U whenever A U and U is open in X. Definition 2.14[8]: Let f be a mapping from (X, ) into (Y, ).Then f is said to be bipolar intuitionistic fuzzy G𝛼 continuous if

is BIF G𝛼CS in (X, ) for every CS in (Y, ). Definition 2.15[8]: Let f be a mapping from (X, ) into (Y, ).Then f is said to be bipolar intuitionistic fuzzy G𝛼 irresolute if

is BIFG𝛼OS in X for every BIFG𝛼OS in X.

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III. BIPOLAR INTUITONISTIC FUZZY G CONNECTED SPACES

Definition 3.1: Two Bipolar intuitionistic fuzzy sets A and B are said to be q-coincident AqB if and only if there exists an element x X such that (x)> (x), (x)< (x), (x)< (x), (x)> (x). Definition 3.2: Two Bipolar intuitionistic fuzzy sets A and B are said to be not q-coincident A B if and only if A . Definition 3.3: A bipolar intuitionistic fuzzy topological space (X, ) is bipolar intutionistic fuzzy-G𝛼 disconnected if

there exists an intuitionistic fuzzy-G𝛼 open sets A,B in X,A 0 ,B 0 such that A B=1 and

A B=0 .If X is not BIFG𝛼 disconnected then it is said to bipolar intuitionistic fuzzy G𝛼-connected. Example 3.4: Let X={a,b}, ={0 } where G={<a,0.3,-0.1,0.1,-0.1>,<b,0.4,-0.1,0.1,-0.1>}.Let A={<a,0.6,-0.1,0.1,-0.1>,<b,0.1,-0.1,0.2,-0.1>} and B={<a,0.8,-0.1,0.1,-0.1>,<b,0.7,-0.1,0.1,-0.1>}. A and B are BIFG𝛼-open sets in X,A 0 ,B and A B=B 1 ,A B=A 0 .Hence X is BIFG𝛼- connected. Example 3.5:Let X={a,b}, ={0 } where G={<a,0.1,-0.1,0.3,-0.1>,<b,0.1,-0.1,0.4,-0.1>}.Let A={<a,0,0,1,-1>,<b,1,-1,0,0>} and B={<a,1,-1,0,0>,<b,0,0,1,-1>}.A and BBIFG𝛼-open sets in

X,A 0 ,B and A B=1 ,A B=0 .Hence X is BIFG𝛼- disconnected. Definition 3.6: Let N be a BIFS in BIFTS (X, )

(i) If there exists BIFG𝛼-open sets M and W in X satisfying the following properties.Then N is called BIFG𝛼 -disconnected(i=1,2,3,4)

N M .

N M .

N M .

N M .

(ii) N is said to be BIFG𝛼 -connected(i=1,2,3,4) if N is notBIFG𝛼 disconnected(i=1,2,3,4).

Obviously we can obtain the following implication between several types of BIFG𝛼 -connectedness(i=1,2,3,4)

BIFG - BIFG -

BIFG -connected

BIFG -connected

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Example 3.7:Let X={a,b}, ={0 } where G={<a,0.3,-0.2,0.3,-0.3>,<b,0.3,-0.3,0.3,-0.3>}.Let M={<a,0.3,-0.4,0.3,-0.3>,<b,0.3,-0.5,0.3,-0.3>} and W={<a,0.3,-0.6,0.3,-0.3>,<b,0.3,-0.7,0.3,-0.3>}. Here M and N are BIFG𝛼-open sets in X. Consider the BIFS N={<a,0.1,-0.3, 0.8,-0.7>,<b,0.2,-0.3,0.8,-0.7>}. N is BIFG𝛼 -connected,BIFG𝛼 -connected,BIFG𝛼 -connected but BIFG𝛼 -disconnected. Example 3.8:Let X={a,b}, ={0 } where G={<a,0.3,-0.3,0.3,-0.5>,<b,0.3,-0.3,0.3,-0.6>}.Let M={<a,0.3,-0.3,0.3,-0.7>,<b,0.3,-0.1,0.3,-0.3>} and W={<a,0,-0.2,0.3,-0.3>,<b,0,-0.3,0.3,-0.3>}.Here M and N are BIFG𝛼-open sets in X.Consider the BIFS N={<a,0.2,0,1,-1>,<b,0.3,0,1,-1>}.N is

BIFG𝛼 -connected but BIFG𝛼 -disconnected. Example 3.9:Let X={a,b}, ={0 } where G={<a,0.1,-0.3,0.3,-0.3>,<b,0.2,-0.3,0.3,-0.3>}.Let M={<a,0.3,-0.3,0.3,-0.3>,<b,0.4,-0.3,0.5,-0.3>} and W={<a,0.5,-0.3,0.3,-0.4>,<b,0.6,-0.3,0.1,-0.3>}.Here M and N are BIFG𝛼-open sets in X.Consider the BIFS N={<a,0.3,-0.3, 0.7,-0.6>,<b,0.5,-0.3,0.5,-0.7>}.N

is BIFG𝛼 -connected,BIFG𝛼 -connected,BIFG𝛼 -connected but BIFG𝛼 -disconnected. Definition 3.10:A BIFTS(X, ) is BIFG𝛼 -disconnected if there exists BIFS A in X,which is both

BIFG𝛼OS and BIFG𝛼CS such that A 0 and A 1 .If X is not BIFG𝛼 -disconnected then it is said to

be BIFG𝛼 -connected. Example 3.11: Let X={a,b}, ={0 } where G={<a,0.1,-0.1,0.7,-0.1>,<b,0.1,-0.1,0.8,-0.1>}.Let A={<a,0.9,-0.1,0.1,-0.1>,<b,0.2,-0.2,0.2,-0.1>}.Here A is BIFG𝛼-open and BIFG𝛼-closed set in X,Hence there exist BIFS in X such that 1 A 0 which is both BIFG𝛼-open and BIFG𝛼-closed set in X. Thus

X is BIFG𝛼 -disconnected. Example 3.12: Let X={a,b}, ={0 } where G={<a,0.1,-0.1,0.3,-0.1>,<b,0.1,-0.1,0.4,-0.1>}.Let A={<a,0.1,-0.1,0.5,-0.1>,<b,0.1,-0.1,0.6,-0.1>}.Here A is BIFG𝛼-open but A is not BIFG𝛼-closed set in X

and 1 A 0 . Thus X is BIFG𝛼 -connected. Proposition 3.13:EveryBIFG𝛼 connected space is BIFG𝛼-connected. Proof: Suppose that there exists non-empty BIFG𝛼-open sets A and B such that A B=1 and

A B=0 (BIFG𝛼-disconnected) then =1 ,

=0 and =0, ,

= -1. In other words, =A.Hence A is an BIFG𝛼-clopen which

implies X is BIFG𝛼 disconnected . Remark 3.14:The converse of the above theorem need not be true as shown in the example below. Example 3.15: Let X={a,b}, ={0 } where G={<a,0.2,-0.2,0.2,-0.6>,<b,0.2,-0.2,0.2,-0.7>}.Let A={<a,0.2,-0.2,0.2,-0.4>,<b,0.2,-0.2,0.2,-0.5>} and B={<a,0.2,-0.2,0.2,-0.2>,<b,0.2,-0.2,0.2,-0.3>}.Here A and B are BIFG𝛼-open sets in X.Also 1 A B={<a,0.2,-0.2,0.2,-0.2>,<b,0.2,-0.2,0.2,-

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0.3>},0 A B={<a,0.2,-0.2,0.2,-0.4>,<b,0.2,-0.2,0.2,-0.5>}.Hence X is BIFG𝛼-connected.Since A is both BIFG𝛼-open and BIFG𝛼-closed set in X, X is BIFG𝛼 -disconnected. Proposition 3.16:Let f:(X, ) (Y, ) be a bipolar intuitionistic fuzzy G𝛼-irresolute surjection.If (X, ) is

bipolar intuitionistic fuzzy G𝛼-connected then (Y, ) is bipolar intuitionistic fuzzy G𝛼-connected. Proof:Assume that (Y, ) is not bipolar intuitionistic fuzzy G𝛼-connected.Then there exists non-empty bipolar intuitionistic fuzzy G𝛼-open sets A and B in (Y, ) such that A B=1 and A B=0 .Since f is a

bipolar intuitionistic fuzzy G𝛼-irresolute mapping,C= 0 ,D= which are bipolar intuitionistic fuzzy G𝛼-open sets in X. =1 which implies C D=1 . =0 which implies C D=0 .Thus X is BIFG𝛼-disconnected,which is a contradiction to our assumption.Hence Y is bipolar intuitionistic fuzzy G𝛼-connected. Theorem 3.17: (X, ) isBIFG𝛼 -connected iff there exists no-nonempty bipolar intuitionistic fuzzy G𝛼-open sets A and B in X such that A= . Proof:Let (X, ) beBIFG𝛼 -connected.Suppose that A and B are bipolar intuitionistic fuzzy G𝛼-open sets in X such that A 0 B and A= Since A= , is an BIFG𝛼OS and B is an BIFG𝛼CS and A which implies B 1 .But this is a contradiction to the fact that X isBIFG𝛼 -connected. Conversely,let A be both bipolar intuitionistic fuzzy G𝛼-open set and bipolar intuitionistic fuzzy G𝛼-closed set in X such that Now take B= .B is a bipolar intuitionistic fuzzy -open set and A which implies B= which implies B which is a contradiction to our hypothesis.Therefore (X, ) is BIFG𝛼 -connected. Theorem 3.18: (X, ) isBIFG𝛼 -connected iff there exists no-nonempty bipolar intuitionistic fuzzy G𝛼-open sets A and B in X such that A= ,B=( and A=( . Proof:Let (X, ) be BIFG𝛼 -connected.Suppose that A and B are bipolar intuitionistic fuzzy G𝛼-open sets in X such that A 0 B,A= ,B=( and A=( .Since ( and ( are BIFG𝛼OS,A and B are BIFG𝛼OS in (X, ).This implies that (X, ) is not a connected space,which is a contradiction.Therefore there exists no -nonempty bipolar intuitionistic fuzzy G𝛼-open sets A and B in X such that A= ,B=( and A=( . Conversely,let A be a both bipolar intuitionistic fuzzy G𝛼-open set and bipolar intuitionistic fuzzy G𝛼-closed set in X such that Now take B= .B is a bipolar intuitionistic fuzzy -open set and A which implies B= which implies B which is a contradiction to our hypothesis.Therefore (X, ) is BIFG𝛼 -connected. Definition 3.19:A BIFTS (X, ) is said to be BIFG𝛼-strongly connected if there exists no non-empty BIFG𝛼CS A and B in X such that

.

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Or in other words a BIFTS (X, ) is called bipolar intuitionistic fuzzy G𝛼-strongly connected if there exists no non-empty bipolar intuitionistic fuzzy G𝛼-closed sets A and B such that A B=0 . Example 3.20: Let X={a,b}, ={0 } where G={<a,0.1,-0.3,0.3,-0.3>,<b,0.2,-0.3,0.3,-0.3>}.Let A={<a,0.3,-0.3,0.3,-0.3>,<b,0.4,-0.3,0.3,-0.3>} and B={<a,0.5,-0.3,0.3,-0.3>,<b,0.6,-0.3,0.3,-0.3>}.A and B are BIFG𝛼CS in X.Hence X is BIFG𝛼-strongly connected . Proposition 3.21:Let f:(X, ) (Y, ) be a bipolar intuitionistic fuzzy G𝛼-irresolute surjection.If (X, ) is a BIFG𝛼-strongly connected then so is Y.

Proof: Let f:(X, ) (Y, ) be a bipolar intuitionistic fuzzy G𝛼-irresolute surjection.If (X, ) is a BIFG𝛼-strongly connected.Suppose Y is not BIFG𝛼-strongly connected then there exists bipolar intuitionistic

fuzzy G𝛼-closed set C and D in Y such that C .Since f is bipolar intuitionistic fuzzy G𝛼-irresolute we have are bipolar intuitionistic fuzzy G𝛼-closed sets in X and

=0 , If then ) C=0 .Hence (X, ) is a BIFG𝛼-super disconnected which is a contradiction to our

hypothesis.Therefore (Y, ) isBIFG𝛼-strongly connected. Remark 3.22:BIFG𝛼 –strongly connected and BIFG𝛼 - connected spaces are independent of each other. Example 3.23:Let X={a,b}, ={0 } where G={<a,0.7,-0.3,0.3,-0.3>,<b,0.3,-0.3,-0.1,-0.3>}.Let A={<a,0.3,-0.7,0.6,-0.3>,<b,0.7,-0.8,0.3,-0.2>} and B={<a,0.3,-0.3,0.2,-0.3>,<b,0.3,-0.3,0.6,-0.3>}.A and B are BIFG𝛼OS in X. Hence X is BIFG𝛼-strongly connected but not BIFG𝛼 connected space since B is BIFG OS and BIFG CS in X. Example 3.24: Let X={a,b}, ={0 } where G={<a,0.2,-0.1,0.4,-0.4>,<b,0.2,-0.2,0.4,-0.4>}.Let A={<a,0.1,-0.2,0.9,-0.8>,<b,0.2,-0.3,0.5,-0.7>} and B={<a,0.3,-0.4,0.6,-0.2>,<b,0.4,-0.6,0.6,-0.3>}.A and B are BIFG𝛼OS in X.Hence X is BIFG𝛼 connected space but not BIFG𝛼-strongly connected such that

.

Definition 3.25:A and B are non-zero BIFS in (X, ).Then A and B are said to be (i)BIFG𝛼-weakly separated if G𝛼cl(A) and G𝛼cl(B) . (ii)BIFG𝛼-q-separated if G𝛼cl(A) B=0 ,A G𝛼cl(B)=0 . Definition 3.26:A BIFTS(X, ) is said to be BIFG𝛼 -disconnected if there exists BIFG𝛼-weakly separated sets and non-zero BIFS, A and B in (X, ) such that A B=1 . Example 3.27:Let X={a,b}, ={0 ,1 } where G={<a,0.3,-0.3,0.3,-0.1>,<b,0.3,-0.3,0.3,-0.2>} .Let A={<a,0,0,1,-1>,<b,1,-1,0,0>} and B={<a,1,-1,0,0>,<b,0,0,1,-1>}.Here A and B are BIFG𝛼OS in X and

G𝛼cl(A) and G𝛼cl(B) .Hence A and B areBIFG𝛼-weakly separated and A B=1 .So (X, ) is BIFG𝛼 -disconnected.

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Definition 3.28:A BIFTS (X, ) is said to be BIFG𝛼 -disconnected if there exists BIFG𝛼-q- separated sets and non-zero BIFS, A and B in (X, ) such that A B=1 . Example 3.29: Let X={a,b}, ={0 ,1 } where G={<a,0.3,-0.3,0.3,-0.1>,<b,0.3,-0.3,0.3,-0.2>}.Let A={<a,0,0,1,-1>,<b,1,-1,0,0>} and B={<a,1,-1,0,0>,<b,0,0,1,-1>}.Here A and B areBIFG𝛼OS in X and G𝛼cl(A) B=0 ,A G𝛼cl(B)=0 . Hence A and B areBIFG𝛼-q-separated and A B=1 .So (X, ) is

BIFG𝛼 -disconnected. Remark 3.30:A BIFTS(X, ) is said to be BIFG𝛼 -connected if and only if (X, ) is BIFG𝛼 –connected. Definition 3.31:A BIFS,A in (X, ) is said to be BIFG𝛼-regular open set if G𝛼int(G𝛼cl(A))=A and BIFG𝛼-regular closed set if G𝛼cl(G𝛼int(A))=A. Definition 3.32:A BIFTS (X, ) is said to beBIFG𝛼-super disconnected if there exists a BIFG𝛼-regular open set A in X such that 0 X is called BIFG𝛼-super connected if X is not BIFG𝛼-super disconnected. Example 3.33:Let X={a,b} and ={0 } where G={<a, 0.3,-0.3,0.3,-0.1>,<b,0.3,-0.3,0.3,-0.2>}.Let A={<a,0,0,1,-1>,<b,1,-1,0,0>} and B={<a,1,-1,0,0>,<b,0,0,1,-1>}.Here A and B are BIFG𝛼OS in X and G𝛼int(G𝛼cl(A))=A.Therefore A is BIFG𝛼-regular open set in X.Hence (X, ) is said to be BIFG𝛼-super disconnected space. Proposition 3.34:Let (X, ) be a BIFTS.Then the following are equivalent:

(i) X is a BIFG𝛼-super connected.

(ii) For each BIFG𝛼OS A in X,we have G𝛼cl(A)=1 .

(iii)For each BIFGCS A in X,we have G𝛼int(A)=0 .

(iv) There exists no BIFG𝛼OS A and B in X such that A B and A .

(v) There exists no BIFG𝛼OS A and B in X such that A B,B= and

A

(vi) There exists no BIFG𝛼CS A and B in X such that A B, B= and

A

Proof: (i) (ii)Let X be a BIFG𝛼-super connected. Assume that there exists A 0 such that G𝛼cl(A) 1 .Take G𝛼int(G𝛼cl(A))=A.Then A is a proper G𝛼-regular open set in X which contradicts that X is BIFG𝛼-super connected. (ii) Let A be a BIFG𝛼 closed set in X.If we take B= then B is a BIFG𝛼-open set in x and B .Hence by (ii) G𝛼cl(B)=1 =0 G𝛼int( )=0 G𝛼int(A)=0 .

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(iii) (iv)Let A and B be BIFG𝛼OS in X such that A B and A .Since is an BIFG𝛼CS in X, 1 .By (iii) G𝛼int( )=0 .But A 0 A=G𝛼int(A) G𝛼int( )=0 which is a contradiction. (iv) (i)Let 0 1 be a BIFG𝛼-regular open set in X.If we take B= we get B

.If not B then =0 =1 A=G =G int(1 )=1 A=1 which is a contradiction since 1 .Also we have A which is also a

contradiction.Therefore X is a BIFG𝛼-super connected. (i) (v)Let A and B be two BIFG OS in (X, ) such that A , B= and

A Now we have G int(G )=G int( )= =A,A and A .Since if

A=1 then 1 = =0 B=0 .But B and A implies that a is a proper BIFG -regular open set in (X, ) which is a contradiction to (i).Hence (v) is true. (v) (i)Let A be BIFG -open set in Xsuch that A=G int(G cl(A)),0 A .Now take B= .In this case we get B and B is BIFG OS in X and B= and ( = =( =G int(G cl(A))=A.But this is a contradiction to (v).Therefore (X, ) is BIFG𝛼-super connected. (v) (vi)Let A and B be BIFG CS in (X, ) such that A B, B= and

A .Taking C= and D= ,C and C are BIFG OS in (X, ) and

C , = = =G = =D and similarly we have =C.But this is a contradiction to (v).Hence (vi) is true.

(vi) Let A and B be BIFG𝛼OS in X such that A B,B= and

A Taking C= and D= ,C and C are BIFG CS in (X, ) and

C . = = =G = =D and similarly we have =C.But this is a contradiction to (vi).Hence (v) is true.

Proposition 3.35: Let f:(X, ) (Y, ) be a BIFG -irresolute surjection.If X is BIFG -super connected then so is Y. Proof:Let X is BIFG -super connected.Suppose Y is not BIFG -super connected.Then there exists BIFG OS C and D in Y such that C ,C Since f is BIFG -irresolute, (C) and (D) are BIFG OS in X and C implies, (C) ( )= .Hence (C) ( ) which means X is BIFG -super disconnected which is a contradiction. Defintion 3.36:A BIFTS(X, ) is called BIFG -connected between two bipolar intuitionistic fuzzy sets A and B if there is no BIFG OS E in (X, ) such that A E and E B. Example 3.37:Let X={a,b}, ={0 ,1 ,G} where G={<a,0.2,-0.2,0.2,-0.6>,<b,0.2,-0.2,0.2,-0.7>}.Let A={<a,0.3,-0.3,0.3,-0.1>,<b,0.3,-0.3,0.3,-0.2>} and B={<a,0.3,-0.3,0.3,-0.3>,<b,0.3,-0.3,0.3,-0.4>} and D={<a,0.2,-0.2,0.2,-0.8>,<b,0.2,0.2,0.2,-0.2>}.Here D is BIFOS in X.Here A D and D .Hence X is BIFG -connected between A and B.

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Definition 3.38:A BIFTS(X, ) is called BIF -connected between two bipolar intuitionistic fuzzy sets A and B if there is no BIFOS, E in (X, ) such that A E and E B. Theorem 3.39:If a BIFTS (X, ) is BIFG -connected between two BIFS,A and B then it is BIF -connected between two BIFS, A and B but not converesely. Proof:Let (X, ) be BIFG -connected between two BIFS,A and B.Suppose (X, ) is not BIF -connected between two BIFS, A and B,then there exists a BIFOS,E in (X, ) such that A E and E B.Since every BIFOS is BIFG OS we have there exists a BIFG OS,E in (X, ) such that A E and E B which implies (X, ) is not BIFG -connected between two BIFS,A and B,a contradiction to our hypothesis.Thus (X, ) is BIF -connected between two BIFS,A and B. Example 3.40: Let X={a,b}, ={0 ,1 ,G} where G={<a,0.2,-0.3,0.3,-0.1>,<b,0.2,-0.3,0.3, -0.2>}. Then (X, ) is BIF -connected between two BIFS’s A={<a,0.2,-0.1,0.8,-0.9>,<b,0.3,-0.2,0.7,-0.8>} and B={<a,0.6,-0.7,0.4,-0.3>,<b,0.5,-0.6,0.5,-0.4>}.But (X, ) is not BIFG -connected between A and B since the BIFS B={<a,0.2,-0.3,0.3,-0.3>,<b,0.2,-0.2,0.3,-0.3>} is BIFG OS such that A E and E . Theorem 3.41:Let (X, ) be a BIFTS and A and B be BIFS in (X, ).If AqB then (X, ) is BIFG -connected between A and B but not converesely. Proof: Let AqB and suppose (X, ) is not BIFG -connected between A and B.Then there exists BIFG OS,E in (X, ) such that A E and E .This implies that A .This implies that A .That is A B which is a contradiction to your hypothesis.Therefore (X, ) is BIFG -connected between A and B. Example 3.42:Let X={a,b}, ={0 ,1 ,G} where G={<a,0.2,-0.2,0.2,-0.6>,<b,0.2,-0.2,0.2,-0.7>}.Let A={<a,0.1,-0.2,0.3,-0.3>,<b,0.1,-0.6,0.3,-0.3>} and B={<a,0.1,-0.2,0.9,-0.8>,<b,0.1,-0.1,0.9,-0.8>} and D={<a,0.2,-0.7,0.2,-0.2>,<b,0.2,0.8,0.2,-0.2>}.Here D is BIFOS in X.Here A D and D .Hence X is BIFG -connected between A and B.But A is not q-coincident with B since

(x)< (x), (x)> (x), (x)> (x), (x)< (x). Theorem 3.43:A BIFTS (X, ) is BIFG -connected between two BIFS A and B iff there is no BIFG OS and BIFG CS, D in (X, ) such that A D . Proof:Let (X, ) is BIFG -connected between two BIFS A and B.Suppose there is BIFG OS and BIFG CS D in (X, ) such that A D ,then A D and D B.This implies that (X, ) is not BIFG -connected between two BIFS A and B which is a contradiction to our hypothesis.Therefore there is no BIFG OS and BIFG CS D in (X, ) such that A D . Conversely, let there exists no BIFG OS and BIFG CS, D in (X, ) such that A D .Suppose (X, ) is not BIFG -connected between two BIFS A and B then there exists BIFG OS,D in (X, ) such that A D and D B.This implies that there is a BIFG OS,D in (X, ) such that A D .But this is a contradiction to our assumption.Therefore (X, ) is BIFG -connected between two BIFS A and B.

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IV. Bipolar intuitionistic fuzzy G -extremally disconnectedness in bipolar intuitionistic fuzzy topological spaces

Definition 4.1:Let (X, ) be a BIFTS.Then X is called bipolar intuitioinistic fuzzy G -extremally disconnected if the G -closure of every BIFG OS is BIFG OS. Theorem 4.2:For a BIFTS (X, ) the following conditions are equivalent.

(i) (X, ) is a BIFG -extremally disconnected space.

(ii) For each BIFG CS A, G int(A) is an BIFG CS.

(iii)For each BIFG OS A, G cl(A)=(

(iv) For each BIFG OS A and B with G cl(A)= , G cl(A)= .

Proof: (i) (ii)Let (X, ) is a BIFG -extremally disconnected space.Let A be a BIFCS.Then is BIFG OS.So BIFG cl( )= is a BIFG OS.Thus is a BIFG CS in (X, ). (ii) (iii)Let A be a BIFG OS.Then ( )= .

= .Since A is BIFG OS, is a BIFG CS.So by (ii) is a BIFG CS.That is = .Hence = (A). (iii) (iv)Let A and B be any two BIFG OS in (X, ) such that G cl(A)= .By (iii), G cl(A)=

= = . (iv) (i)Let A be BIFG OS in (X, ).Put B= .Then G cl(A)= Hence by (iv),G cl(A)=

.Therefore G cl(A) is BIFG OS in (X, ).That is (X, ) is extremally disconnected space.

V. CONCLUSIONS

Connectedness in bipolar intuitionistic fuzzy environment is introduced and discussed.Further the concept is extended to BIFG connectedness.Various forms of connectedness and their inter relationships with each other are also studied and counter examples for the same are listed.

ACKNOWLEDGMENT

I would place on record my sincere note of thanks to my guide Dr.Rev.Sr.M.Helen for her timely suggestions and inputs which helped me in each step in the construction of the paper.

REFERENCES [1] Atanassov K.T., “Intuitionistic fuzzy sets”, Fuzzy Sets and Systems, 20(1986), 87-96.

[2] Chang C.L., “Fuzzy topological spaces”, J Math. Anal. Appl., 24(1968), 182-190.

[3] Chiranjibe Jana, Madhumangal Pal, “Application of Bipolar Intuitionistic Fuzzy Soft Sets in Decision Making Problem”, International Journal of

Fuzzy System Applications,7(3) ,32-55, 2018.

[4] Dogan Coker, “An introduction to intuitionistic fuzzy topological spaces”, Fuzzy Sets and Systems. 88(1997),81-89.

[5] Ezhilmaran.D & Sankar K. (2015), “Morphism of bipolar intuitionistic fuzzy graphs”, Journal of Discrete Mathematical Sciences and Cryptography,

18(5), 605-621.

[6] Lee. K.M. 2000, “Bipolar-valued fuzzy sets and their operations”, Proc. Int. Conf. on Intelligent Technologies, Bangkok,Thailand, 307-312.

[7] Ludi Jancy Jenifer.K,Helen.M, “Bipolar intuitionistic fuzzy generalised alpha closed sets via bipolar intuitionistic fuzzy topological

spaces”,International Journal of Research and Analytical Reviews,6(2)(2019),642-648.

[8] Ludi Jancy Jenifer.K,Helen.M, “Bipolar intuitionistic fuzzy generalised alpha continuous and irresolute functions in bipolar intuitionistic fuzzy

environment”,(communicated).

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[9] S. Ozcag and D. Coker, “ On connectedness in intuitionistic fuzzy special topological spaces”, Inter. J.Math. Sci. 21 (1998) 33-40.

[10] Zadeh L.A., “ Fuzzy Sets”, Information and Control, 8(1965), 338-353.

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